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There is sadly widespread confusion and misunderstanding about how physicists work. The aim of Physics is to describe nature by constructing mathematical models. This sounds scary, but a mathematical model is just a set of equations that allow us to make predictions about what happens in the real world. All mathematical models are approximate and have limited accuracy, but we can in principle at least make our models as accurate as we want by working harder at them.

Anyhow, the point of all this is that the Big Bang arises from a particular solution of the equations of General Relativity called the FLRW metric. General Relativity is a mathematical model, and almost certainly an approximate one since we expect it to be modified by some future theory of quantum gravity. However GR has made correct predicitions every time it's been tested, so we think it's an extremely accurate model when applied to the universe around us, and we have a lot of confidence in it.

GR is exceedingly complex and hard to solve, so if we want to use GR to describe the whole universe we have to make some approximations. Specifically these are that the universe is homogeneous and isotropic. Homogeneity means the universe is the same everywhere, and isotropic means it looks the same in all directions. At small scales these assumptions obviously aren't true because starts are clumped into galaxies so the distribution of matter is uneven. However galaxies are small compared to the visible universe, and when you look at length scales of a billion light years the density of matter averages out and is roughly homogeneous. So we expect a theory based on isotropy and homogeneity to be accurate at large scales but less so at smaller scales.

Anyhow, if we make these assumptions it turns out to be fairly easy to solve the equations of GR for the whole universe and the solution is the FLRW metric that I mentioned earlier. The interesting thing about this solution is that it predicts the universe must be expanding (or contracting): it can't stand still. When we look around we find that the universe is indeed expanding, and in fact the FLRW solution appears to be an excellent description of the universe at very large scales.

So, we believe GR, and we believe the FLRW metric (for large scales), and if we use the FLRW metric to extrapolate back in time we find it predicts that 13.7 billion years ago the universe shrank to nothing. This point is what we call the Big Bang. It may seem a stretch to say that because the FLRW metric correctly describes the universe today it can be relied upon to correctly describe the universe 13.7 billion years ago. However we can see back in time because the farther away galaxies are the longer light took to reach us. The oldest galaxies we can see date from less than a billion years after the Big Bang. Plus we can measure the microwave radiation from an event called recombination that occurred only 300,000 years after the Big Bang (you've probably heard of this radiation - it's called the cosmic microwave background). All these observations match the calculations made using the FLRW metric.

So the bottom line is that we believe the FLRW metric is a good description of the universe, because it matches experiment, and the FLRW metric predicts the Big Bang, so that's why we believe the Big Bang happened.

Well, we are not sure about that, we know things like that the universe is expanding with accelerated speed, plus many other different observable properties, in the same time we build different mathematical models of the universe putting some "intuitive" assumptions, and then we simulate in computers how our universe will look like (evolve in time) in those models, and the standard Big Bang theory is just one of many others, and it seems that it predicts our current observable picture of the universe relatively good, anyway there is many fundamental problems with all this models, what makes physics actually fun ;)